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Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.
If the operation is multiplicative then the elements are g0, g1, g2, ...
Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
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Is every finite abelian group is cyclic?
No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic
What is the order of a group?
The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.
What is the order of an element of a group?
The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.
The set of all counting numbers less than one million is it finite or infinite?
Finite.
What is the difference of finite and infinite?
its 2 types of numbers Finite numbers: have an end. they can be extremely long, but as long as they have an end at some point they are finite Infinite numbers never end. 1/3 is infinite, because the 0,3333 continues forever without even reaching exactly 1/3. 1/4 is finite, because it is exactly 0,25. So basically infinite numbers can never be written down exactly (in a decimal way), no matter how much paper you use, whereas finite numbers can.
